Between Traditional Schooling and Homeschooling: Criteria for the Legitimation of Mathematical Learning According to Its Actors

Abstract

The study examines the criteria through which mathematical learning is recognized and validated in traditional schooling and homeschooling. Unlike research focused on academic performance, it examines the narratives of students, parents/guardians, and teachers alongside mathematical productions from modeling tasks linked to the function object, in order to characterize what counts as evidence of learning, what is valued, who validates it, and through which procedures it is legitimized in each modality.

The study adopts an interpretive qualitative approach through a multiple case study design, involving four triads —two per modality— comprising a student, a parent/guardian, and a teacher; in the homeschooling modality, the mathematics teacher provides support remotely. Data sources included narrative logs, follow-up tasks, an initial diagnostic task, a closing task, students' written productions, and research team observations.

The theoretical framework, articulated synergistically, draws on Legitimation Code Theory, Bernstein's sociology of educational knowledge, Sfard's commognitive theory, and, in a complementary role, Lakoff and Johnson's conceptual metaphor theory, organized around five dimensions: authorized actors, valued knowledge, accepted evidence, validation procedures, and attributed purposes.

Findings show that traditional schooling legitimizes learning through the curriculum, teacher planning, and teacher authority, whereas homeschooling distributes validation among the parent/guardian, the online teacher, digital resources, and external certification. Both modalities share difficulties in legitimizing more complex forms of mathematical learning, such as argumentation and graphical interpretation. The difference between modalities does not lie in performance, but in the criteria through which learning is validated.